A099985 -id:A099985 - cp's OEIS frontend

A204455 Squarefree product of all odd primes dividing n, and 1 if n is a power of 2: A099985/2.

1, 1, 3, 1, 5, 3, 7, 1, 3, 5, 11, 3, 13, 7, 15, 1, 17, 3, 19, 5, 21, 11, 23, 3, 5, 13, 3, 7, 29, 15, 31, 1, 33, 17, 35, 3, 37, 19, 39, 5, 41, 21, 43, 11, 15, 23, 47, 3, 7, 5, 51, 13, 53, 3, 55, 7, 57, 29, 59, 15, 61, 31, 21, 1, 65, 33, 67, 17, 69, 35, 71, 3

Offset: 1

Wolfdieter Lang, Jan 19 2012

There are no odd primes dividing n iff n is a power of 2.
This sequence coincides with the bisection of A007947 (even indices), which is A099985, dividing out the even prime 2 in the squarefree kernel.
a(n) divides A106609(n) for n>=1. - Alexander R. Povolotsky, Apr 06 2015

			a(5)=5 because 5 is a single odd prime.
a(9)=3 because 9=3*3 has as squarefree part 3.
a(1)=1 because 1 is a power of 2, having no odd primes as a factor.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A099985, A099984, A007947, A000265, A065463, A106609.

A204455 := proc(n)
    local p;
    numtheory[factorset](n) minus {2} ;
    mul(p,p=%) ;
end proc:
seq(A204455(n),n=1..40) ; # R. J. Mathar, Jan 25 2017

f[n_] := Select[First /@ FactorInteger@ n, PrimeQ@ # && OddQ@ # &]; Times @@@ (f /@ Range@ 120) (* Michael De Vlieger, Apr 08 2015 *)

a(n) = {my(f = factor(n)); prod(k=1, #f~, if (f[k,1] % 2, f[k,1], 1));} \\ Michel Marcus, Apr 07 2015

a(n) = factorback(setminus(factorint(n)[, 1]~, [2])) \\ Jianing Song, Aug 09 2022

a(n) = A099985(n)/2 = A007947(2*n)/2.
a(n) = A000265(A007947(n)) = A007947(A000265(n)). - Charles R Greathouse IV, Jan 19 2012
Multiplicative with a(p^e)=p for p <> 2 and a(2^e)=1. - R. J. Mathar, Jul 02 2013
a(n) = Sum_{d|n} phi(d)*mu(2d)^2. - Ridouane Oudra, Sep 02 2019
From Richard L. Ollerton, May 09 2021: (Start)
a(n) = Sum_{k=1..n} mu(2*n/gcd(n,k))^2.
a(n) = Sum_{k=1..n} mu(2*gcd(n,k))^2*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = (2/5) * Product_{p prime} (1 - 1/(p*(p+1))) = (2/5) * A065463 = 0.281776... . - Amiram Eldar, Nov 19 2022
a(n) = Sum_{d divides n, d odd} mu(d)^2 * phi(d). - Peter Bala, Feb 01 2024

A007947 Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 2, 65, 66, 67, 34, 69, 70, 71, 6, 73, 74, 15, 38, 77, 78

Offset: 1

R. Muller, Mar 15 1996

Multiplicative with a(p^e) = p.
Product of the distinct prime factors of n.
a(k)=k for k=squarefree numbers A005117. - Lekraj Beedassy, Sep 05 2006
A note on square roots of numbers: we can write sqrt(n) = b*sqrt(c) where c is squarefree. Then b = A000188(n) is the "inner square root" of n, c = A007913(n), b*c = A019554(n) = "outer square root" of n, and a(n) = lcm(a(b),c). Unless n is biquadrateful (A046101), a(n) = lcm(b,c). [Edited by Jeppe Stig Nielsen, Oct 10 2021, and Andrey Zabolotskiy, Feb 12 2025]
a(n) = A128651(A129132(n-1) + 2) for n > 1. - Reinhard Zumkeller, Mar 30 2007
Also the least common multiple of the prime factors of n. - Peter Luschny, Mar 22 2011
The Mobius transform of the sequence generates the sequence of absolute values of A097945. - R. J. Mathar, Apr 04 2011
Appears to be the period length of k^n mod n. For example, n^12 mod 12 has period 6, repeating 1,4,9,4,1,0, so a(12)= 6. - Gary Detlefs, Apr 14 2013
a(n) differs from A014963(n) when n is a term of A024619. - Eric Desbiaux, Mar 24 2014
a(n) is also the smallest base (also termed radix) for which the representation of 1/n is of finite length.  For example a(12) = 6 and 1/12 in base 6 is 0.03, which is of finite length. - Lee A. Newberg, Jul 27 2016
a(n) is also the divisor k of n such that d(k) = 2^omega(n). a(n) is also the smallest divisor u of n such that n divides u^n. - Juri-Stepan Gerasimov, Apr 06 2017

			G.f. = x + 2*x^2 + 3*x^3 + 2*x^4 + 5*x^5 + 6*x^6 + 7*x^7 + 2*x^8 + 3*x^9 + ... - _Michael Somos_, Jul 15 2018

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
Masum Billal, Divisible Sequence and its Characteristic Sequence, arXiv:1501.00609 [math.NT], 2015, theorem 11 page 5.
Henry Bottomley, Some Smarandache-type multiplicative sequences
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Jarosław Grytczuk, Thue type problems for graphs, points and numbers, Discrete Math., 308 (2008), 4419-4429.
Neville Holmes, Integer Sequences [Broken link]
Serge Lang, Old and New Conjectured Diophantine Inequalities, Bull. Amer. Math. Soc., 23 (1990), 37-75. see p. 39.
Wolfdieter Lang, Cantor's List of Real Algebraic Numbers of Heights 1 to 7, arXiv:2307.10645 [math.NT], 2023.
D. H. Lehmer, Euler constants for arithmetical progressions, Collection of articles in memory of Juriĭ Vladimirovič Linnik. Acta Arith. 27 (1975), 125--142. MR0369233 (51 #5468). See N_k on page 131.
Ivar Peterson, The Amazing ABC Conjecture
Paul Tarau, Emulating Primality with Multiset Representations of Natural Numbers, in Theoretical Aspects of Computing, ICTAC 2011, Lecture Notes in Computer Science, 2011, Volume 6916/2011, 218-238
Paul Tarau, Towards a generic view of primality through multiset decompositions of natural numbers, Theoretical Computer Science, Volume 537, 5 June 2014, Pages 105-124.
Wikipedia, Radical of an integer.

See A007913, A062953, A000188, A019554, A003557, A066503, A087207 for other properties related to square and squarefree divisors of n.
More general factorization-related properties, specific to n: A020639, A028234, A020500, A010051, A284318, A000005, A001221, A005361, A034444, A014963, A128651, A267116.
Range of values is A005117.
Cf. A048803, A019565, A024619, A053462, A060735, A073355 (partial sums), A079277, A097945, A129132, A225546.
Bisections: A099984, A099985.
Sequences about numbers that have the same squarefree kernel: A065642, array A284311 (A284457).
A003961, A059896 are used to express relationship between terms of this sequence.
Cf. A001615, A008683, A076479, A078615.

a007947 = product . a027748_row  -- Reinhard Zumkeller, Feb 27 2012

[ &*PrimeDivisors(n): n in [1..100] ]; // Klaus Brockhaus, Dec 04 2008

with(numtheory); A007947 := proc(n) local i,t1,t2; t1 := ifactors(n)[2]; t2 := mul(t1[i][1],i=1..nops(t1)); end;
A007947 := n -> ilcm(op(numtheory[factorset](n))):
seq(A007947(i),i=1..69); # Peter Luschny, Mar 22 2011
A:= n -> convert(numtheory:-factorset(n),`*`):
seq(A(n),n=1..100); # Robert Israel, Aug 10 2014
seq(NumberTheory:-Radical(n), n = 1..78); # Peter Luschny, Jul 20 2021

rad[n_] := Times @@ (First@# & /@ FactorInteger@ n); Array[rad, 78] (* Robert G. Wilson v, Aug 29 2012 *)
Table[Last[Select[Divisors[n],SquareFreeQ]],{n,100}] (* Harvey P. Dale, Jul 14 2014 *)
a[ n_] := If[ n < 1, 0, Sum[ EulerPhi[d] Abs @ MoebiusMu[d], {d, Divisors[ n]}]]; (* Michael Somos, Jul 15 2018 *)
Table[Product[p, {p, Select[Divisors[n], PrimeQ]}], {n, 1, 100}] (* Vaclav Kotesovec, May 20 2020 *)

a(n) = factorback(factorint(n)[,1]); \\ Andrew Lelechenko, May 09 2014

for(n=1, 100, print1(direuler(p=2, n, (1 + p*X - X)/(1 - X))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020

from sympy import primefactors, prod
def a(n): return 1 if n < 2 else prod(primefactors(n))
[a(n) for n in range(1, 51)]  # Indranil Ghosh, Apr 16 2017

def A007947(n): return mul(p for p in prime_divisors(n))
[A007947(n) for n in (1..60)] # Peter Luschny, Mar 07 2017

(define (A007947 n) (if (= 1 n) n (* (A020639 n) (A007947 (A028234 n))))) ;; ;; Needs also code from A020639 and A028234. - Antti Karttunen, Jun 18 2017

If n = Product_j (p_j^k_j) where p_j are distinct primes, then a(n) = Product_j (p_j).
a(n) = Product_{k=1..A001221(n)} A027748(n,k). - Reinhard Zumkeller, Aug 27 2011
Dirichlet g.f.: zeta(s)*Product_{primes p} (1+p^(1-s)-p^(-s)). - R. J. Mathar, Jan 21 2012
a(n) = Sum_{d|n} phi(d) * mu(d)^2 = Sum_{d|n} |A097945(d)|. - Enrique Pérez Herrero, Apr 23 2012
a(n) = Product_{d|n} d^moebius(n/d) (see Billal link). - Michel Marcus, Jan 06 2015
a(n) = n/( Sum_{k=1..n} (floor(k^n/n)-floor((k^n - 1)/n)) ) = e^(Sum_{k=2..n} (floor(n/k) - floor((n-1)/k))*A010051(k)*M(k)) where M(n) is the Mangoldt function. - Anthony Browne, Jun 17 2016
a(n) = n/A003557(n). - Juri-Stepan Gerasimov, Apr 07 2017
G.f.: Sum_{k>=1} phi(k)*mu(k)^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Apr 11 2017
From Antti Karttunen, Jun 18 2017: (Start)
a(1) = 1; for n > 1, a(n) = A020639(n) * a(A028234(n)).
a(n) = A019565(A087207(n)). (End)
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s)). - Vaclav Kotesovec, Dec 18 2019
From Peter Munn, Jan 01 2020: (Start)
a(A059896(n,k)) = A059896(a(n), a(k)).
a(A003961(n)) = A003961(a(n)).
a(n^2) = a(n).
a(A225546(n)) = A019565(A267116(n)). (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = A065463/2. - Vaclav Kotesovec, Jun 24 2020
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))^2.
a(n) = Sum_{k=1..n} mu(gcd(n,k))^2*phi(gcd(n,k))/phi(n/gcd(n,k)).
For n>1, Sum_{k=1..n} a(gcd(n,k))*mu(a(gcd(n,k)))*phi(gcd(n,k))/gcd(n,k) = 0.
For n>1, Sum_{k=1..n} a(n/gcd(n,k))*mu(a(n/gcd(n,k)))*phi(gcd(n,k))*gcd(n,k) = 0. (End)
a(n) = (-1)^omega(n) * Sum_{d|n} mu(d)*psi(d), where omega = A001221 and psi = A001615. - Ridouane Oudra, Aug 01 2025

More terms from several people including David W. Wilson

Definition expanded by Jonathan Sondow, Apr 26 2013

A099984 Bisection of A007947.

Original entry on oeis.org

1, 3, 5, 7, 3, 11, 13, 15, 17, 19, 21, 23, 5, 3, 29, 31, 33, 35, 37, 39, 41, 43, 15, 47, 7, 51, 53, 55, 57, 59, 61, 21, 65, 67, 69, 71, 73, 15, 77, 79, 3, 83, 85, 87, 89, 91, 93, 95, 97, 33, 101, 103, 105, 107, 109, 111, 113, 115, 39, 119, 11, 123, 5, 127, 129, 131, 133, 15, 137

Offset: 1

N. J. A. Sloane, Nov 19 2004

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007947, A065463, A099985.

with(numtheory): A007947 := proc(n) local i,t1,t2; t1 :=ifactors(n)[2]; t2 := mul(t1[i][1],i=1..nops(t1)); end: seq(A007947(2*n-1),n=1..78); # Emeric Deutsch, Dec 15 2004

a[n_] := Times @@ (First /@ FactorInteger[2*n-1]); Array[a, 100]  (* Amiram Eldar, Nov 19 2022*)

a(n) = factorback(factorint(2*n-1)[, 1]); \\ Amiram Eldar, Nov 19 2022

From Amiram Eldar, Nov 19 2022: (Start)
a(n) = A007947(2*n-1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (6/5) * Product_{p prime} (1 - 1/(p*(p+1))) = (6/5) * A065463 = 0.8453306... . (End)

More terms from Emeric Deutsch, Dec 15 2004

Offset corrected by Amiram Eldar, Nov 19 2022

A374436 Triangle read by rows: T(n, k) = Product_{p in PF(n) union PF(k)} p, where PF(a) is the set of the prime factors of a.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 6, 3, 2, 2, 2, 6, 2, 5, 5, 10, 15, 10, 5, 6, 6, 6, 6, 6, 30, 6, 7, 7, 14, 21, 14, 35, 42, 7, 2, 2, 2, 6, 2, 10, 6, 14, 2, 3, 3, 6, 3, 6, 15, 6, 21, 6, 3, 10, 10, 10, 30, 10, 10, 30, 70, 10, 30, 10, 11, 11, 22, 33, 22, 55, 66, 77, 22, 33, 110, 11

Offset: 0

Peter Luschny, Jul 10 2024

			  [ 0]  1;
  [ 1]  1,  1;
  [ 2]  2,  2,  2;
  [ 3]  3,  3,  6,  3;
  [ 4]  2,  2,  2,  6,  2;
  [ 5]  5,  5, 10, 15, 10,  5;
  [ 6]  6,  6,  6,  6,  6, 30,  6;
  [ 7]  7,  7, 14, 21, 14, 35, 42,  7;
  [ 8]  2,  2,  2,  6,  2, 10,  6, 14,  2;
  [ 9]  3,  3,  6,  3,  6, 15,  6, 21,  6,  3;
  [10] 10, 10, 10, 30, 10, 10, 30, 70, 10, 30,  10;
  [11] 11, 11, 22, 33, 22, 55, 66, 77, 22, 33, 110, 11;

Family: A374433 (intersection), A374434 (symmetric difference), A374435 (difference), this sequence (union).

Cf. A007947 (column 0, main diagonal), A099985 (central terms).

PF := n -> ifelse(n = 0, {}, NumberTheory:-PrimeFactors(n)):
A374436 := (n, k) -> mul(PF(n) union PF(k)):
seq(print(seq(A374436(n, k), k = 0..n)), n = 0..11);

nn = 12; Do[Set[s[i], FactorInteger[i][[All, 1]]], {i, 0, nn}]; s[0] = {1}; Table[Apply[Times, Union[s[k], s[n]]], {n, 0, nn}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jul 11 2024 *)

# Function A374436 defined in A374433.
for n in range(12): print([A374436(n, k) for k in range(n + 1)])

T(0,0) = T(n,0) = 1; T(n,k) = rad(k*n) where rad = A007947. - Michael De Vlieger, Jul 11 2024

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A204455 Squarefree product of all odd primes dividing n, and 1 if n is a power of 2: A099985/2.

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A007947 Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.

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A099984 Bisection of A007947.

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A374436 Triangle read by rows: T(n, k) = Product_{p in PF(n) union PF(k)} p, where PF(a) is the set of the prime factors of a.

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